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'' $  M $
of a [[Group|group]]  $  G $
in a subgroup  $  H $
of  $  G $''

The set

$$ 
N _ {H} ( M)  = \ 
\{ {h } : {h \in H , h  ^ {-} 1 M h = M } \}
,
$$

that is, the set of all elements  $  h $
of  $  H $
such that  $  h  ^ {-} 1 m h $(
the conjugate of  $  m $
by  $  h $) 
for every  $  m \in M $
also belongs to  $  M $. 
For any  $  M $
and  $  H $
the normalizer  $  N _ {H} ( M) $
is a subgroup of  $  H $. 
An important special case is the normalizer of a subgroup of a group  $  G $
in  $  G $. 
A subgroup  $  A $
of a group  $  G $
is normal (or invariant, cf. [[Invariant subgroup|Invariant subgroup]]) in  $  G $
if and only if  $  N _ {G} ( A) = G $. 
The normalizer of a set consisting of a single element is the same as its [[Centralizer|centralizer]]. For any  $  H $
and  $  M $
the cardinality of the class of subsets conjugate to  $  M $
by elements of  $  H $(
that is, subsets of the form  $  h  ^ {-} 1 M h $, 
$  h \in H $) 
is equal to the index  $  | H :  N _ {H} ( M) | $.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M.I. Kargapolov,   J.I. [Yu.I. Merzlyakov] Merzljakov,   "Fundamentals of the theory of groups" , Springer  (1979)  (Translated from Russian)</TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D.J.S. Robinson,   "A course in the theory of groups" , Springer  (1980)</TD></TR></table>